contributions from the theory of conceptual fields: help and feedback messages in educational...

8/9/2019 Contributions From The Theory Of Conceptual Fields: Help And Feedback Messages In Educational Software For De

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ICME 11 TSG 22 THEME NUMBER 3

CONTRIBUTIONS FROM THE THEORY OF

CONCEPTUAL FIELDS: HELP AND FEEDBACK MESSAGES

IN EDUCATIONAL SOFTWARE FOR DEAF STUDENTS

Maici Duarte Leite [email protected]

Rute Elizabete de Souza Rosa Borba [email protected]

Alex Sandro Gomes [email protected]

Universidade Federal de Pernambuco, Brazil

AbstractThis paper consists of the report and analysis of an empirical research developed by the first author as a Masters

dissertation and supervised by the third and second authors. The aim of the study was to develop an interface design

composed with additive problems to be used by both hearing and deaf students. Through competitor analysis,observation of software usage, specialists assessments and final users testing, a prototype was developed. Based on

Vergnauds Theory of Conceptual Fields, problems were presented that varied in operation meanings (combine,

compare, change and equalize problems) and that used different representational supports to aid the understanding of

each particular situation. Deaf students preferred the use of diagrams to help them overcome their difficulties thatwere basically relational but that had also numerical components. Diagrams were useful because pupils could rely on

visual information to understand the main problem information and, in particular, the relations involved and what

was being asked. The developed prototype was, thus, successful in aiding deaf students in overcoming difficulties in

their understanding of addition and subtraction problems by visually providing useful help and feedback messages

and also making it possible for these pupils to interact through technology with their hearing pairs.

Introduction

Students difficulties in the acquisition of mathematical knowledge have been considered

in educational proposals and many of these suggest the use of technology as means to overcome

problems in understanding and to provide ways of motivating pupils in their conceptual

development.

Difficulties in mathematical understanding are even greater amongst deaf students.

Previous studies (Austin, 1975; Traxler, 2000; Nunes and Moreno, 2002; Kelly, Lang and

Pagliaro, 2003a; Kelly, Lang, Mousley and Davis, 2003b; Bull, Marschark and Vallee, 2005;

Zarfaty, Nunes and Bryant, 2004) have shown that deaf students are far behind in mathematical

development when compared with hearing colleagues. The main reasons appointed for this gap in

cognitive development are associations between linguistic and experiential factors, because

language is developed differently amongst deaf and amongst hearing pupils and social

experiences are also distinct between these two groups.Although sign language has been recognized as an alternative to oral expression, in order

to minimize deaf students difficulties, the use of this language in social spaces and also in

educational software is still restricted. Technology can be used to mediate deaf students learning

processes but there are not sufficient adequate products for these pupils, especially in

mathematics education.

The present study proposed an interface design composed with additive problems. The

methodology used was user-centered, that is, incorporated users needs, in particular those of deaf

students but also took in consideration that classroom must include students with and without

special needs and, thus, software should be adequate for both hearing and for deaf pupils.

The design process proposed involved: competitor analysis, observation of software

usage, specialists assessments and final users testing, resulting in a prototype that aims to attenddeaf students specificities and limitations in an inclusive context. The investigation aimed to

observe how natural language and icons of sign language (LIBRAS Brazilian Sign Language)


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used in help and feedback messages, appropriate for each type of additive problem, can aid

hearing and deaf pupils in their understanding of addition and subtraction and that could also be

transferred to any other interface.

Deaf pupils learning of mathematics

Most studies that involve deaf pupils point out that these students learning difficulties are

basically related to language acquisition and that this affects their social relationships andcognitive development.

Bull, Marschark e Vallee (2005) reinforce that deaf pupils low performance in

mathematics is directly related to cultural factors and language acquisition because the lack of

coordination between linguistic, symbolic and analogical forms of representation of number can

lead to difficulties in the learning of arithmetic concepts. Hearing pupils, on the other hand, learn

about correlations using visual and audible information, as when relating an object with the oral

expression used to designate it. Deaf pupils cannot rely on audible information so this must be

considered in their learning, in particular of mathematics.

Nunes and Moreno (2002) argue that even very basic mathematical concepts such as

simple additions and subtractions require the coordination between action schemes and systems

of signs culturally developed. Thus, if the coordination of words and other symbols is necessary

in the understanding of numbers and arithmetic operations, then deaf students are in

disadvantage.

Teaching of mathematics has, thus, to consider the particularities of deaf students

learning processes and proposals must aim to overcome these pupils difficulties and to include

them in classroom interactions with hearing pairs. Educational software may be designed as a

resource with these aims in mind.

A theoretical approach to conceptual development

The main theory used as reference in the design proposed in the present study was that of

Grard Vergnaud the Theory of Conceptual Fields in which vast sets of situations andconcepts are considered in an articulated manner as means of understanding how knowledge is

acquired and developed. According to Vergnaud (1986), a concept is built based on a triplet of

three sets: Situations (S), Invariants (I) and Representations (R). The analysis of problem-

situation is central in this theory, and also the study of procedures and symbolic representations

used by pupils whilst discussing, arguing and solving problems (Vergnaud, 1997).

Additive problems, according to this theory, involve, amongst other concepts, addition

and subtraction problems that vary in meanings involved, in properties and relations implicit

and explicit in problem solving and in symbolic means of representing these concepts.

Additive problems were chosen for the present study because of their wide usage in many

school levels and fundamental for pupils mathematical development, both for hearing and for

deaf students.Additive problems, that involve additions and/or subtractions, isolated or in combination,

of natural or directed numbers, were classified by Vergnaud (1991) in six categories: composition

of measures; transformation applied to a measure; comparison of measures; composition of two

transformations; composition of relations; and transformation applied to a relation.

Another classification of additive problems was proposed by Carpenter and Moser (1982)

that explores only natural numbers, distributed in four categories: combine, compare, change and

equalize. In combine problems parts are joined to compose unities; in compare problems

measures are compared; in change problems a transformation is applied to an initial measure

resulting in a new measure; and in equalize problems two measures are to be transformed in order

to be the same in quantity. These categories have many similarities with the classification

proposed by Vergnaud and consider that additive problems have many different meanings. Thesefour categories result in 16 distinct situations, depending on where the unknown value is situated.


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Because only natural numbers are involved, in the present study Carpenter and Mosers

classification was used because the software was aimed for primary school pupils that do not deal

formally with directed numbers.

This classification structures the problems that were part of the software proposed in this

paper and was used as basis of the diagrams designed as help andfeedbackmessages and also to

set the level of difficulties of the problems to be proposed to the users. Deaf students particular

difficulties were considered, as well as the belief that technology can be used to explore a varietyof problem-situations and, in this manner, enrich these pupils cognitive experiences, in particular

in the additive field, by offering diagrams as representational supports.

User-centered design

User-centered design considers users abilities, needs, expectations, goals and tasks to be

performed (Hall, 2001). The understanding of users needs and objectives is required, generating

a product that is adequate and useful for the user, although much information about the users

needs to be collected and interpreted (Preece, Rogers e Sharp, 2005).

In the present study, by means of a user-centered design, the following techniques were

applied: competitor analysis, observation and analysis of software usage, specialist evaluation

and testing by final users. These techniques resulted in a prototype that aimed to attend specific

needs and limitations of deaf users, in an inclusive context. The steps followed were those listed

below and may be schematically observed in Figure 1:

Deaf pupils needs were identified concerning mathematical knowledgeacquisition, by means of literature review;

Five available software with focus on additive structures were selected andanalyzed with the objective of identifying aspects that could be incorporated;

Six deaf students were observed in the usage of these software, identifying theirrepresentational needs and relations between concepts;

A first prototype was developed that was assessed by a mathematics educatorspecialist and a specialist in LIBRAS (Brazilian Sign Language) and theseanalyses lead to a new prototype version;

The new prototype version was piloted and this lead to prototype modifications; This version was tested by users, generating other alterations and The final version of the prototype was consolidated.

Prototype4 release

Test usability (2)

OF

C

U

Prototype3 release

Test usability(1)

OFC

U

Prototype2 release

Prototype1 release

Analysis of competitors

User observation

Requirements:- Mathematical

- Interface design

Assessmentby experts

Figure 1 Flow chart of techniques used

This process resulted in the listing of requirements of three types: adequate for any

educational software, specific to deaf students and specific for the study of the additiveconceptual field.


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Results: requirements identified

General requirements

These apply to software designed for both hearing and deaf pupils, such as:

Privileging the use of motivational resources; Allowing the user to return to executed actions, revisiting and reflecting on actions before

obtaining final answers;

Making possible the access to different levels of difficulty, respecting distinct levels ofusers knowledge;

Aiding comprehension by means of feedback that lead the users to reflect aboutmathematical contents;

Offering the possibility for users to identify themselves and emitting messages with thisidentification

Making available reports with the activities developed and performance of users so theteacher can better aid pupil development;

Allowing users to follow up their own performance; Permitting teachers to configure and include new activities with other levels of difficulty.

Requirements specific to deaf pupils

These requirements make it possible for the interface to attend deaf students

communication needs and can be integrated to an interface to also be used by hearing pupils:

Exploring legends, animations, movements and other visual effects, such as colorcontrasts, and also short and clear texts, verbs in the infinitive and messages in graphic

forms;

Representing LIBRAS signs; Exploring visual alerts to direct user actions in an interactive manner; Presenting a screen simulation for understanding of users rules; Presenting activities, help andfeedbackmessages in LIBRAS.

Requirements specific to explore the additive field

For both hearing and deaf pupils to explore additive situations:

Offering varied help options in accordance to problem type Presenting different representations of additive structures Following up the type of error: relational (in the choice of procedures, strategies and

operations) or numerical (related to calculations performed)

Exploring the positional system of numbers in the final answers, because deaf pupilssometimes present difficulties in registering groups of tens and units in a conventional

manner.

Results: help and feedback messages explored

In order to attend specific needs of the deaf students both help and feedbackmessages

were given in written Portuguese and by use of icons inLIBRAS (Brazilian Language of Signs).

One of the requirements observed was that help options need to vary, in order to

contemplate the educational proposal of the software (in this case the understanding of different

types of additive problems). Help options need to consider experience and knowledge levels of

the users and must also be compatible with the exploration of the distinct additive categories. The

help offered must, thus, be associated to the different types of problems.

In the case of this study, distinct help was offered for the problems, according to the

classification proposed by Carpenter and Moser (1982), for combination, for comparison, forchange and for equalization problems. Three different types of help were offered: metaphor of


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concrete material, algorithm and diagram, as may be observed in Figure 2. The student could,

thus, choose the form of symbolic representation that would better aid the understanding of the

problem structure and meanings of the operations involved. The first form allowed users to

virtually manipulate the objects involved in problem-situations. The second symbolic support

allowed users to structure their reasoning in terms of addition and subtraction algorithms and the

third form allowed users to access diagrams that represented each particular situation (combine,

compare, change and equalize problems).

Figure 2. The different types of help provided: metaphor of concrete material, algorithm and

diagrams.

It was observed that in the case of the deaf students that tested the prototype, diagrams

were the preferred form of help because the schematic representation represented the situation

without having to rely exclusively on written words. The diagrams highlighted key invariants

(relations and properties specific to a certain class of problems), aiding the user to go beyond

trial-and-error tentative.

Feedback messages were also given in accordance to the type of additive problem being

solved. For example, if in a change problem the student added given values instead of

subtracting, a feedback message questioned if it was possible to obtain, in that particular case, a

greater value as a result.

This integration of help and feedback messages were, thus, organized according to

problem categories, considering relational and numerical errors, i.e., errors in the choice of

operation or in the calculus of the result. Feedback messages did not simply tell students they had

obtained an incorrect answer but lead them to think about their solutions.

Relational errors referred to the incomprehension of the implicit relations of the problem

structure. Appropriate feedback messages were given to each incorrect choice of problem solving

strategy, thus, a message for a combination problem was not the same for a comparison problem,

because each involves distinct invariants.

Numerical errors produced specific messages according to types of error: incorrect

additions or subtractions an/or incorrect use of terms (such as change in minuends andsubtrahends).


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In combine problems (diagram in the upper left side of Figure 2) the diagram

representation highlighted the static relation between quantities and their subparts. A thin line

separated one characters quantity involved in the situation to the other characters quantity and

the circle represented the total amount made up by joining these separate quantities. This diagram

aided the comprehension that wholes are made up of subparts and that the whole may be

determined if subparts are known or that a subpart may be determined if the whole and other

subparts are known. Feedback was given in accordance to error type:

TOGETHER THEY HAVE LESS? (in case of relational error) ADD AGAIN (in case of numerical error)

Students were, thus, questioned if it was possible to obtain a smaller number than the ones

added or were asked to repeat their addition if a correct value had been obtained.

Compare problems (diagram in the upper right side of Figure 2) involved static relations

and the diagram allowed pupils to observe the difference between quantities. The two characters

had the same quantity in an equal recipient and the difference was represented out of the

recipient. Feedback messages given in errors of these cases were:

THE GIRL HAS LESS THAN THE BOY? (in case of relational error) ADD AGAIN (in case of numerical error)If the error was of a relational nature, pupils were questioned about how they should

proceed to obtain the difference between the greater and the smaller quantities and if of a

numerical nature they were asked to repeat their calculation.

The diagram of change problems (lower left side of Figure 2) represented a dynamic

relation in which an initial quantity by a direct action is increased or decreased. Color contrasts

were used to highlight that a change had occurred. Feedback messages given in case of relational

and numerical errors were, respectively:

THE GIRL GAVE ORANGES TO THE BOY. HOW CAN SHE HAVEMORE?

SUBTRACT AGAINThe pupils were encouraged to reflect on what would be the value at the end of an action

(of decrease or increase) or how they had incorrectly been operated (subtracted or added).

Finally, in equalize problems the diagram (lower right side of Figure 2) represented a

dynamic relation in which quantities must be made equal. The scale metaphor helped children in

comparing quantities and determining the exceeding value (to be increased to the lower value or

decreased to the larger one). Feedback messages were:

ARE QUANTITIES EQUAL? (in case of relational error) SUBRACT AGAIN (in case of numerical error).

Students were in this manner encouraged to think about the actions required to make

quantities equal or were asked to repeat their calculation when the choice of operation had been

correct but the result obtained was incorrect by calculation error or by change of the position of

minuends and subtrahends.

The testing of the prototype showed that different strategies were used by problem type

and, thus, specific help and feedback messages were useful in helping students understand the

different relations involved in each type of problem. This variety also allowed pupils to have a

broad contact with the additive conceptual field.

Conclusion

The analysis of the whole process of elaborating, testing and assessing prototypes proved

to be successful in designing a means to aid both deaf and hearing pupils in their difficulties in

solving addition and subtraction problems.

Having a broad and robust theoretical basis such as the one proposed by VergnaudsTheory of Conceptual Fields offered conditions for the development of a prototype that

considered mathematical aspects (such as difference in meanings for operations) and also


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students cognition processes (of both hearing and deaf pupils). The classification of additive

problems used took in consideration the subjacent relations present in addition and subtraction

problems and help and feedback messages were based on the diversity of additive situations.

These messages not only alerted students that they had obtained an incorrect answer as occurs

frequently in educational software but also gave them elements to reflect on the possible causes

of their errors. Differentiating between numerical and relational difficulties was very important

because pupils were encouraged to think on the specificity of their errors. Representationalsupport was also a key issue to the success of the prototype developed because different forms of

symbolism were provided that helped students to look at the problems from different perspectives

and aided specially deaf pupils in providing visual supports for understanding the problems and

not relying excessively in written representation.

The diversity of additive situations, the representational support and the help and

feedback messages provided were very useful for deaf pupils and certainly may aid hearing

students in their understanding of addition and subtraction problems and, in a general way,

contribute to the reflecting on how help and feedback messages can be presented in educational

software.

This paper is, thus, believed to bring contributions to studies on the use of technologies in

the teaching and learning of mathematics, to the research of cognitive processes focused in

mathematics education, in a broader sense, and, specifically, to investigations on deaf pupils

learning processes. The present study points out necessary aspects of each of these fields of

research and also highlights the need to develop more interdisciplinary studies that investigate

how technology can be used to help overcome difficulties in mathematics that can be caused by

physical particularities, such as those of deaf pupils.

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